+26387 what is the area of A(0,3) B(4,-1) C(-2,1) D(2,-3)
\(\begin{array}{|rcll|} \hline \text{Let } \vec{A} &=& \binom{0}{3} \\ \text{Let } \vec{B} &=& \binom{4}{-1} \\ \text{Let } \vec{C} &=& \binom{-2}{1} \\ \text{Let } \vec{D} &=& \binom{2}{-3} \\ \hline \end{array}\)
Points clockwise:
\(\begin{array}{|rcll|} \hline 1. & \vec{A} &=& \binom{0}{3} \\ 2. & \vec{B} &=& \binom{4}{-1} \\ 3. & \vec{D} &=& \binom{2}{-3} \\ 4. & \vec{C} &=& \binom{-2}{1} \\ \hline \end{array}\)
Area:
\(\begin{array}{|rcll|} \hline 2A &=& |\vec{B} \times \vec{A}| + |\vec{D} \times \vec{B}| + |\vec{C} \times \vec{D}| + |\vec{A} \times \vec{C}| \\ 2A &=& |\binom{4}{-1} \times \binom{0}{3}| + |\binom{2}{-3} \times \binom{4}{-1}| + |\binom{-2}{1} \times \binom{2}{-3}| + |\binom{0}{3} \times \binom{-2}{1}| \\ 2A &=& 4\cdot 3-(-1)\cdot 0 +2(-1)-(-3)\cdot 4 +(-2)(-3)-1\cdot2 + 0\cdot 1 -3(-2) \\ 2A &=& 12-0 -2+12 +6-2 + 0 +6 \\ 2A &=& 32 \\ \mathbf{A} & \mathbf{=}& \mathbf{16} \\ \hline \end{array}\)
+129839 This forms a rectangle
AB = √ [ (4 - 0 )^2 + ( - 1 - 3)^2] = √[4^2 + (-4)^2 ] = √[16 + 16] = √ 32 = 4√2
BD = √ [ (4 - 2 )^2 + ( - 3 - - 1)^2] = √[2^2 + (-2)^2 ] = √[4 + 4] = √ 8 = 2√2
So...the area = 4√2 * 2√2 = 8 * 2 = 16 sq units
+26387 what is the area of A(0,3) B(4,-1) C(-2,1) D(2,-3)
\(\begin{array}{|rcll|} \hline \text{Let } \vec{A} &=& \binom{0}{3} \\ \text{Let } \vec{B} &=& \binom{4}{-1} \\ \text{Let } \vec{C} &=& \binom{-2}{1} \\ \text{Let } \vec{D} &=& \binom{2}{-3} \\ \hline \end{array}\)
Points clockwise:
\(\begin{array}{|rcll|} \hline 1. & \vec{A} &=& \binom{0}{3} \\ 2. & \vec{B} &=& \binom{4}{-1} \\ 3. & \vec{D} &=& \binom{2}{-3} \\ 4. & \vec{C} &=& \binom{-2}{1} \\ \hline \end{array}\)
Area:
\(\begin{array}{|rcll|} \hline 2A &=& |\vec{B} \times \vec{A}| + |\vec{D} \times \vec{B}| + |\vec{C} \times \vec{D}| + |\vec{A} \times \vec{C}| \\ 2A &=& |\binom{4}{-1} \times \binom{0}{3}| + |\binom{2}{-3} \times \binom{4}{-1}| + |\binom{-2}{1} \times \binom{2}{-3}| + |\binom{0}{3} \times \binom{-2}{1}| \\ 2A &=& 4\cdot 3-(-1)\cdot 0 +2(-1)-(-3)\cdot 4 +(-2)(-3)-1\cdot2 + 0\cdot 1 -3(-2) \\ 2A &=& 12-0 -2+12 +6-2 + 0 +6 \\ 2A &=& 32 \\ \mathbf{A} & \mathbf{=}& \mathbf{16} \\ \hline \end{array}\)
